On the Smith-Thom deficiency of Hilbert squares

We give an expression for the Smith-Thom deficiency of the Hilbert square \(X^{[2]}\) of a smooth real algebraic variety \(X\) in terms of the rank of a suitable Mayer-Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of \(X^{[2]}\) in the case of projective complete intersections, and show that with a few exceptions no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.